Crystal balls are in short supply,
especially in the stock market. Yet every investor wants to know the growth
rate of future earnings per share (EPS). While the norms of a company's past
EPS growth are no sure indicator of the future, they do lend perspective.

If you decide to calculate these
norms, it is important to do it correctly. Otherwise, bogus results will warp
your frame of reference. Here are the procedures to use and the pitfalls to
avoid when you compute growth-rate norms.

* * *

How
to begin.Collect a company's annual EPS data for an
extended time—say, seven to ten years. Arrange the EPS in chronological order.
All EPS must be positive, as in this ten-year sequence from a large southeastern
bank.

$2.25, $2.47, $2.76, $3.13, $3.04, $3.50, $4.30, $4.70,
$4.66, $4.73

Make
ratios of the sequence's adjacent EPS, putting the earlier EPS in the ratio's
denominator and the later EPS in its numerator, like this.

The ratio values are annualgrowth factors for the EPS. A growth factor always equals 1+g.
The number 1 stands for 100% of the earlier EPS. And g is the
EPS growth rate—it tells us the proportion of the earlier EPS (the denominator)
by which the later EPS (the numerator) increased or decreased.

Since the growth factor is 1+g,
the growth rate g is the growth factor minus 1. Here are two
growth-rate examples from the above data.

Once we have calculated the EPS growth
factors, we can examine their year-to-year volatility—an important thing to
look into. But more significantly, we can determine their norms; specifically,
(1) the growth factors' mean (i.e., their average value) and (2) the
growth factors' standard deviation (i.e., the degree to which they are
scattered around their mean).

A
pitfall: the arithmetic mean and standard deviation.There
are different kinds of means. The most familiar is the arithmetic mean.
To calculate the arithmetic mean of n growth factors, we add them and
divide their sum by n, as follows:

(1.0978+1.1174+1.1341+.
9712+1.1513+1.2286+1.0930+. 9915+1.0150)/9=
9.7999/9=
1.0889, the arithmetic mean of the growth factors—the quotient of a
nine-number sum divided by nine

The scatter of growth-factor values
around their arithmetic mean is measured by the arithmetic standard deviation.
The labor-intensive computation of the arithmetic standard deviation is programmed
into spreadsheets and inexpensive scientific calculators. Most statistics
books explain the underlying math.

The arithmetic standard deviation
is a quantity. Multiples of that quantity are added to
(or subtracted from) the arithmetic mean to determine the set of values
that lie within a given range of dispersion. To determine the value that lies
n standard deviations from the mean, we must multiply the arithmetic
standard deviation by n.

The above growth factors have an
arithmetic standard deviation of .083. The growth factor that lies, say, two
standard deviations above the mean is 1.2549, which equals the mean plus
two standard deviations: 1.0889 + 2(.083) = 1.2549. The value that lies two
standard deviations below the mean is .9229, which equals the mean minus
two standard deviations: 1.0889 – 2(.083) = .9229.

Later, we'll use these procedures
as stepping-stones to our goal—but only as that. In and of themselves, the
arithmetic mean and standard deviation are inappropriate measures of growth-factor
norms.

The
geometric mean and standard deviation. The appropriate measures
of growth-factor norms are the less-familiar geometric mean and standard deviation.

To
calculate the geometric mean of n growth factors, we multiply the numbers
together and take the nth root of their product, as shown
here.

[(1.0978)(1.1174)(1.1341)(.9712)(1.1513)(1.2286)(1.0930)(.9915)(1.0150)]1/9=
2.10221/9=
1.0861, the geometric mean of the growth factors—the ninth root of
a nine-factor product

We can also find the geometric mean
by using a shortcut: Divide the final year's earnings (EPSn) by the
first year's earnings (EPS0), and take the nth
root of the result. In our banking example, the final EPS is $4.73, the first
EPS is $2.25, and n is nine because the ten EPS span nine year-long periods.
In general, the number of growth factors is one less than the number of EPS.

(EPSn/EPS0)1/n=
($4.73/$2.25)1/9=(2.1022)1/9=
1.0861, the geometric mean of the growth factors

The scatter of growth-factor values
around their geometric mean is measured by the geometric standard deviation.
The geometric standard deviation can be found using a spreadsheet or a scientific
calculator, but first the growth factors must be converted into logarithms (as
explained below).

In contrast to the arithmetic standard
deviation, the geometric standard deviation is not a quantity; it is
a factor. Powers of the geometric standard deviation are multiplied
by (or divided into) the geometric mean to determine the set of values
that lie within a given range of dispersion. To determine the value that lies
n geometric standard deviations from the mean, we must raise the geometric
standard deviation to the nth power.

The above growth factors have a geometric
standard deviation of 1.0795. The growth factor that lies, say, two standard
deviations above the mean is 1.2657—which equals the mean times
the second power of the standard deviation: (1.0861)(1.07952) = 1.2657.
And the value that lies two standard deviations below the mean is .9320—which
equals the mean divided by the second power of the standard deviation:
1.0861/(1.07952) = .9320.

A
comparison of arithmetic and geometric norms.In the
above examples, there is little difference between the growth factors' arithmetic
mean (1.0889) and geometric mean (1.0861). Similarly, there is little difference
in the values that lie two standard deviations above those means—1.2549 (arithmetic)
and 1.2657 (geometric). And little difference in the values that lie two standard
below those means—.9229 (arithmetic) and .9320 (geometric). So why even bother
with the geometric mean and standard deviation?

Because analytical tools must fit
the kind of data they are examining—if they don't, they can generate seriously
distorted results.

The arithmetic mean and arithmetic
standard deviation are sum-based values. As such, they are appropriate for
additive processes. But earnings growth is not additive; it is multiplicative.
The appropriate measures for EPS growth factors are the geometric mean and geometric
standard deviation, which are product-based values.

To drive this point home, we'll apply
both the arithmetic mean and the geometric mean to our bank's EPS sequence and
examine the results. Recall that a ten-EPS sequence has nine growth factors.
The first EPS times the nine growth factors equals the final EPS.

It follows that the first EPS times
nine mean-growth factors should also equal the final EPS. But when we
use the arithmetic mean, 1.0889, this doesn't work out. The result is not the
final EPS of $4.73, but $4.84, which is eleven cents higher.

$2.25[(1.0861)(1.0861)(1.0861)(1.0861)(1.0861)(1.0861)(1.0861)(1.0861)(1.0861)]=
$2.25[1.08619]=
$2.25[2.1029]=
$4.73, the final EPS(Because
our geometric mean is rounded, the value of 1.08619 is slightly
higher than the product of the nine annual growth factors.)

Average
dangers.We now turn to two EPS sequences whose arithmetic
and geometric means have markedly different values.

High
or higher?The first sequence comes from a manufacturer
of medical devices. Here are its EPS in chronological order.

The arithmetic mean of these factors
is 1.2584, implying that EPS grew at an average annual rate of 25.84%. Had
that been so, the first EPS, $0.32, would have grown into $2.53 in the last
year—which, of course, it did not.

EPS0
(1+g)n = EPSn$0.32(1.25849)= $2.53≠
$1.60, the final EPS

The true mean is 1.1958, the geometric
mean. The first EPS, growing at an annual rate of 19.58%, does indeed become
the final EPS, $1.60.

EPS0
(1+g)n = EPSn$0.32(1.19589)= $1.60, the final EPS

If
you accept the arithmetic mean for the norm, you will overstate the average
annual growth rate by 6.26%.

Growing
or shrinking?We end this section with the EPS sequence
of a global communications company, where there is a dramatic difference between
the two means. Here are its EPS in chronological order.

The arithmetic mean of these factors
is 1.7368, implying that EPS grew at an average annual rate of 73.68%. Had
that been so, the first EPS, $1.47, would have grown into a whopping $211.38
in the final year!

EPS0
(1+g)n = EPSn$1.47(1.73689)= $211.38≠
$1.27, the final EPS

The true mean is the geometric mean,
0.9839, which reveals an average annual decrease in EPS of -1.61%. The
first EPS growing (negatively) by that annual factor does indeed become the
final EPS, $1.27.

EPS0
(1+g)n = EPSn$1.47(0.98399)= $1.27, the final EPS

If
you accept the arithmetic mean for the norm, you will think that EPS growth
has been skyrocketing—nothing could be farther from the truth.

A
side trip through the garden of logarithms.By definition,
the standard deviation measures a data set's dispersion around its arithmetic
mean. But we require a standard deviation around its geometric mean.
Are we out of luck? Not at all! We can find the geometric standard deviation
using logarithms. If you are logarithmically challenged, here's a five-paragraph
mini-course.

Substituting
exponential numbers for ordinary numbers.Any given
positive number can be expressed as an exponential power of some other positive
number. The "other" number is called a base and the required
exponent is called the logarithm (or log) of the given number.
For example, 1.995262315 can be expressed as 10.3 because 10.3
equals 1.995262315. Ten is the base, and its exponent .3 is the log of 1.995262315.
(You can find a number's base-ten logarithm using the LOG function of a spreadsheet
or scientific calculator.)

Interpreting
exponents.The meaning of the decimal exponent .3 will
be clearer if you convert it to its fractional equivalent 3/10. The expression
103/10 tells us: "Raise ten to the third power" (103
= 10*10*10 = 1000) "and take the tenth root of the result" (10001/10
= 1.995262315). Alternatively and equivalently, 103/10 tells us:
"Take the tenth root of ten" (101/10 = 1.258925412) "and
raise the result to the third power" (1.2589254123 = 1.995262315).
Either way, 10.3 equals 1.995262315, so the base-ten logarithm of
1.995262315 is .3.

A logarithm is negative when the
positive number it represents is less than 1. For example, the log of .501187233
is –0.3. The negative sign in the exponent of 10-.3 indicates the
reciprocal of 10.3, which is 10.3 divided into
1. Otherwise, the exponent has the same meaning as above. Thus

10
-.3=
1/10.3=
1/1.995262315=
.501187233

Substituting
a log sum for a factor product.The product of two
or more like-based numbers is the sum of their exponents applied to that base.
For example, (10.0406)(10.0481) = 10.0406 + .0481
= 10.0887. Therefore, we can multiply ordinary positive numbers
by: (1) adding their corresponding logarithms (i.e., their base-ten exponents)
and (2) converting the sum back into an ordinary number, like this.

The
product 1.226591639 is the antilog of .0887 (i.e., ten raised to the
power of the logarithm .0887).

Substituting
a log quotient for a product root.The nth
root of an exponential power is the exponent divided by n and applied
to the same base. For example, the ninth root of 10.0774 is 10.0774/9
= 10.0086 = 1.02. Thus, we can find the nth root of
a product by adding its factors' logs, dividing the sum by n, and
converting the quotient to an ordinary number—as here, where we find the square
root of a two-factor product.

Back
to business.Taken in combination, these facts let
us use logarithms to compute the geometric mean and the geometric standard deviation
of a set of growth factors.

First, we add the logs of the
n growth factors and divide the sum by n to get the logs' arithmetic
mean. The arithmetic mean of the logs is the log of the growth factors' geometric
mean.

Then, we find the logs' arithmetic
standard deviation. The arithmetic standard deviation of the logs is the
log of the growth factors' geometric standard deviation.

Finally, we re-express the logs'
arithmetic mean and arithmetic standard deviation as ordinary numbers (i.e.,
we take their antilogs) and arrive at the factors' geometric mean and geometric
standard deviation. In summary:

The antilog of the logs' arithmetic
mean is the geometric mean of the growth factors.

10arithmetic mean of the factors' logs = geometric mean of the
factors

The antilog of the logs' arithmetic standard deviation
is the geometric standard deviation of the growth factors.

Applying
the procedure.Here again are our bank's previously
calculated growth factors, 1+g, along with their respective base-ten
logarithms, LOG (1+g).

1+g

LOG
(1+g)

1.0978

.04052

1.1174

.04821

1.1341

.05465

0.9712

-.01269

1.1513

.06119

1.2286

.08941

1.0930

.03862

0.9915

-.00370

1.0150

.00647

LOGSUM .32268

The
nine logs have a sum of .32268, an arithmetic mean of .03585 (their sum divided
by nine), and an arithmetic standard deviation of .03322 (computed using a spreadsheet
or scientific calculator). The
geometric mean of the factors is ten to the power of the logs' arithmetic
mean—i.e., 10.03585, which equals 1.0861. The
geometric standard deviation of the factors is ten to the power of the logs'
arithmetic standard deviation—i.e., 10.03322, which equals 1.0795.

Final pitfalls. Even
the armor of mathematical precision can't safeguard us from two remaining dangers.

1. The danger that the
norms reflect a skewed data set. Ideally, the EPS data set should encompasses
a complete business cycle. A data set that contains a partial cycle will likely
be biased, over-weighting either the high or the low growth factors.

2. The danger that the norms reflect
the bygone state of a changing business. A business is subject to a variety
of micro- and macroeconomic forces. The norms of EPS growth are byproducts
of these forces. When forces are stable, norms endure—like the wave pattern
at the confluence of merging streams. When forces are in flux, the old norms
give way to the new. Thus we must be vigilant for currents of change. The
predictive power of our calculated norms is determined by the resilience of
the forces that produced them.